| L(s) = 1 | − 51.9·2-s + 655.·4-s − 3.68e3·5-s − 8.37e4·7-s + 7.24e4·8-s + 1.91e5·10-s − 1.32e5·11-s + 1.93e5·13-s + 4.35e6·14-s − 5.10e6·16-s − 2.59e6·17-s − 4.52e6·19-s − 2.41e6·20-s + 6.91e6·22-s − 7.42e6·23-s − 3.52e7·25-s − 1.00e7·26-s − 5.48e7·28-s − 2.05e7·29-s + 4.28e7·31-s + 1.17e8·32-s + 1.35e8·34-s + 3.08e8·35-s − 8.25e7·37-s + 2.35e8·38-s − 2.66e8·40-s + 7.25e8·41-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.319·4-s − 0.527·5-s − 1.88·7-s + 0.781·8-s + 0.605·10-s − 0.248·11-s + 0.144·13-s + 2.16·14-s − 1.21·16-s − 0.443·17-s − 0.418·19-s − 0.168·20-s + 0.286·22-s − 0.240·23-s − 0.722·25-s − 0.166·26-s − 0.602·28-s − 0.185·29-s + 0.269·31-s + 0.617·32-s + 0.509·34-s + 0.992·35-s − 0.195·37-s + 0.481·38-s − 0.411·40-s + 0.978·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 + 51.9T + 2.04e3T^{2} \) |
| 5 | \( 1 + 3.68e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 8.37e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 1.32e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.93e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.59e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 4.52e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 7.42e6T + 9.52e14T^{2} \) |
| 31 | \( 1 - 4.28e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 8.25e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.25e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.14e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 3.13e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.04e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 8.68e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 4.85e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.44e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.67e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.46e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.86e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.74e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.10e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.02e11T + 7.15e21T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.540921073290903219598054168542, −8.797449890503396275455381157988, −7.78485453888162817247495763257, −6.90081354762725293131881891793, −5.96984807883447947250660525726, −4.36904947696955265812892848389, −3.39924553863294254471814096137, −2.18285201609752889706105470875, −0.67985953042574741050951043205, 0,
0.67985953042574741050951043205, 2.18285201609752889706105470875, 3.39924553863294254471814096137, 4.36904947696955265812892848389, 5.96984807883447947250660525726, 6.90081354762725293131881891793, 7.78485453888162817247495763257, 8.797449890503396275455381157988, 9.540921073290903219598054168542
